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Mirrors > Home > MPE Home > Th. List > clwlkcomp | Structured version Visualization version GIF version |
Description: A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.) |
Ref | Expression |
---|---|
isclwlke.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isclwlke.i | ⊢ 𝐼 = (iEdg‘𝐺) |
clwlkcomp.1 | ⊢ 𝐹 = (1st ‘𝑊) |
clwlkcomp.2 | ⊢ 𝑃 = (2nd ‘𝑊) |
Ref | Expression |
---|---|
clwlkcomp | ⊢ ((𝐺 ∈ 𝑋 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (ClWalks‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlkcomp.1 | . . . . . . 7 ⊢ 𝐹 = (1st ‘𝑊) | |
2 | 1 | eqcomi 2832 | . . . . . 6 ⊢ (1st ‘𝑊) = 𝐹 |
3 | clwlkcomp.2 | . . . . . . 7 ⊢ 𝑃 = (2nd ‘𝑊) | |
4 | 3 | eqcomi 2832 | . . . . . 6 ⊢ (2nd ‘𝑊) = 𝑃 |
5 | 2, 4 | pm3.2i 473 | . . . . 5 ⊢ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃) |
6 | eqop 7733 | . . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = 〈𝐹, 𝑃〉 ↔ ((1st ‘𝑊) = 𝐹 ∧ (2nd ‘𝑊) = 𝑃))) | |
7 | 5, 6 | mpbiri 260 | . . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = 〈𝐹, 𝑃〉) |
8 | 7 | eleq1d 2899 | . . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (ClWalks‘𝐺) ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺))) |
9 | df-br 5069 | . . 3 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ 〈𝐹, 𝑃〉 ∈ (ClWalks‘𝐺)) | |
10 | 8, 9 | syl6bbr 291 | . 2 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (ClWalks‘𝐺) ↔ 𝐹(ClWalks‘𝐺)𝑃)) |
11 | isclwlke.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | isclwlke.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
13 | 11, 12 | isclwlke 27560 | . 2 ⊢ (𝐺 ∈ 𝑋 → (𝐹(ClWalks‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))) |
14 | 10, 13 | sylan9bbr 513 | 1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (ClWalks‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 if-wif 1057 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 {csn 4569 {cpr 4571 〈cop 4575 class class class wbr 5068 × cxp 5555 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 0cc0 10539 1c1 10540 + caddc 10542 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 Vtxcvtx 26783 iEdgciedg 26784 ClWalkscclwlks 27553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-wlks 27383 df-clwlks 27554 |
This theorem is referenced by: clwlkcompim 27563 |
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